Question

Graph each system and determine the number of solutions that it has. If it have one solution, name it.

Answer 1

Given the system of equations below:

`\begin{gathered} x+2y=4\text{ ---- equation 1} \\ y=-(1)/(2)x+2---\text{ equation 2} \end{gathered}`

From equation, to graph equation 1, we solve for y for various values of x.

Thus,

`\begin{gathered} when\text{ x =-8,} \\ -8+2y=4 \\ collect\text{ like terms,} \\ 2y=12 \\ divide\text{ both sides by 2,} \\ (2y)/(2)=(12)/(2) \\ \Rightarrow y=6 \\ \\ when\text{ x = 6,} \\ 6+2y=4 \\ collect\text{ like terms,} \\ 2y=-2 \\ divide\text{ both sides by 2,} \\ (2y)/(2)=(-2)/(2) \\ \Rightarrow y=-1 \end{gathered}`

By plotting the values of x and y as points (x, y), we have the graph of equation 1 to be

Similarly, from equation 2, the values of y for various values of x is thus

`\begin{gathered} when\text{ x = 6,} \\ y=-(1)/(2)(6)+2=-1 \\ when\text{ x = -8,} \\ y\text{ = -}(1)/(2)(-8)+2=6 \end{gathered}`

By plotting the values of x and y as points (x, y), we have the graph of equation 2 to be

By combining the graphs, we have infinitely many solutions.